L'accès aux articles de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.centre-mersenne.org/), implique l'accord avec les conditions générales d'utilisation (http://jtnb. centre-mersenne.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.centre-mersenne.org/ Journal de Théorie des Nombres de Bordeaux 33 (2021), 261-271 Quantitative Diophantine approximation with congruence conditions par Mahbub ALAM, Anish GHOSH et Shucheng YU Résumé. Dans ce court article, nous prouvons une version quantitative du théorème de Khintchine-Groshev avec des conditions de congruence. Notre argument repose sur un argument classique de Schmidt sur le comptage de points de réseau génériques, qui à son tour repose sur une certaine borne de variance sur l'espace des réseaux.
The present paper is concerned with equidistribution results for certain flows on homogeneous spaces and related questions in Diophantine approximation. Firstly, we answer in the affirmative, a question raised by Kleinbock, Shi and Weiss [KSW17] regarding equidistribution of orbits of arbitrary lattices under diagonal flows and with respect to unbounded functions. We then consider the problem of Diophantine approximation with respect to rationals in a fixed number field. We prove a number field analogue of a famous result of W. M. Schmidt which counts the number of approximates to Diophantine inequalities for a certain class of approximating functions. Further we prove "spiraling" results for the distribution of approximates of Diophantine inequalities in number fields. This generalizes the work of Athreya, Ghosh and Tseng [AGT15, AGT14] as well as Kleinbock, Shi and Weiss [KSW17].If · is taken to be the supremum norm then c m can be taken to be 1. In [AGT15], Athreya, Ghosh and Tseng considered the problem of 'spiraling' of approximates connected to the Diophantine inequality above. LetGiven A ⊆ S m−1 , T > 0, they considered the counting functions N(x, T ) = #{(p, q) ∈ Z m × Z + : qx − p < c m |q| −1/m , 0 < q ≤ T } and N(x, T, A) = #{(p, q) ∈ Z m × Z + : qx − p < c m |q| −1/m , 0 < q ≤ T, θ(p, q) ∈ A}, and proved: A.
We prove functional central limit theorems for lattice point counting for affine and congruence lattices using the method of moments. Our main tools are higher moment formulae for Siegel transforms on the corresponding homogeneous spaces, which we believe to be of independent interest.
We prove a quantitative theorem for Diophantine approximation by rational points on spheres. Our results are valid for arbitrary unimodular lattices and we further prove ‘spiraling’ results for the direction of approximates. These results are quantitative generalizations of the Khintchine-type theorem on spheres proved in Kleinbock and Merrill (Israel J Math 209:293–322, 2015).
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